category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
The notion of dual morphism is the generalization to arbitrary monoidal categories of the notion of dual linear map in the category Vect of vector spaces.
Given a morphism $f \colon X \to Y$ between two dualizable objects in a monoidal category $(\mathcal{C}, \otimes)$, the corresponding dual morphism
is the one obtained by $f$ by composing the duality unit of $X$ (the coevaluation map), the duality counit of $Y$ (the evaluation map)β¦
This notion is a special case of the the notion of mate in a 2-category.
Namely if $K \coloneqq \mathbf{B}_\otimes \mathcal{C}$ is the delooping 2-category of the monoidal category $(\mathcal{C}, \otimes)$, then objects of $\mathcal{C}$ correspond to morphisms of $K$, dual objects correspond to adjunctions and morphisms in $\mathcal{C}$ correspond to 2-morphisms in $K$. Under this identification a morphism $f \colon X \to Y$ in $\mathcal{C}$ may be depicted as a 2-morphism of the form
and duality on morphisms is then given by the mate bijection
In $\mathcal{C} =$ Vect with its standard tensor product monoidal structure, a dual object is a dual vector space and a dual morphism is a dual linear map.
If $A$, $B$ are C*-algebras which are PoincarΓ© duality algebras, hence dualizable objects in the KK-theory-category, then for $f \colon A \to B$ a morphism it is K-oriented, the corresponding Umkehr map is (postcomposition) with the dual morphism of its opposite algebra version:
See at KK-theory β Push-forward in KK-theory.
More generally, twisted Umkehr maps in generalized cohomology theory are given by dual morphisms in (β,1)-category of (β,1)-modules. See at twisted Umkehr map for more.